Local Maxima of the Potential Energy on Spheres
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Let S d be a unit sphere in ℝ d+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N ∈ S d , we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ i≠j ||x i − x j ||−α . The following theorem is proved: for α ≥ d − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima.
Keywords
Potential Energy Local Maximum Point Theorem Unit Sphere Positive Real Number
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References
- 1.A. Bondarenko and M. Viazovska, “Spherical designs via Brouwer fixed point theorem,” SIAM J. Discrete Math., 24, 207–217 (2010).CrossRefMATHMathSciNetGoogle Scholar
- 2.H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres,” J. Amer. Math. Soc., 20, No. 1, 99–148 (2007).CrossRefMATHMathSciNetGoogle Scholar
- 3.E. B. Saff and A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intelligencer, 19, No. 1, 5–11 (1997).CrossRefMATHMathSciNetGoogle Scholar
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© Springer Science+Business Media New York 2014